Find the surface area of the cylinder below. Use the approximation 3.14 for pi.

To find the surface area of a cylinder, we use the formula:

\[ \text{Surface Area} = 2\pi r(h + r) \]

where \( r \) is the radius and \( h \) is the height.

Given:

• Radius \( r = 5 \) cm
• Height \( h = 15 \) cm

Using the approximation \( \pi \approx 3.14 \):

First, calculate the expression inside the parentheses, \( h + r \):

\[ h + r = 15 , \text{cm} + 5 , \text{cm} = 20 , \text{cm} \]

Now apply the surface area formula:

\[ \text{Surface Area} = 2 \cdot 3.14 \cdot 5 , \text{cm} \cdot 20 , \text{cm} \]

Calculating the multiplication step-by-step:

1. First, calculate \( 2 \cdot 3.14 = 6.28 \).
2. Next, calculate \( 6.28 \cdot 5 = 31.4 \).
3. Finally, calculate \( 31.4 \cdot 20 = 628 \).

Thus, the surface area of the cylinder is:

\[ \text{Surface Area} = 628 , \text{cm}^2 \]

So, the surface area of the cylinder is \( \boxed{628} , \text{cm}^2 \).

Adrian is painting the outside of a cylinder that he plans to use as a coffee table. The cylinder has a radius of 4 feet and a height of 3 feet. Adrian wants to paint all around the outside of the cylinder, including the top and bottom faces. In order to understand how much paint is needed, he wants to know the surface are of the outside of the cylinder. What is the surface area of the cylinder, measured in square feet? Use 3.14 for pi and round your answer to the nearest tenth

To find the total surface area of a cylinder, we can use the formula:

\[ \text{Surface Area} = 2\pi r(h + r) \]

where:

• \( r \) is the radius,
• \( h \) is the height.

Given:

• Radius \( r = 4 \) feet
• Height \( h = 3 \) feet

Using the approximation \( \pi \approx 3.14 \):

1. First, calculate \( h + r \): \[ h + r = 3 , \text{ft} + 4 , \text{ft} = 7 , \text{ft} \]

2. Now, substitute \( r \) and \( h + r \) into the surface area formula: \[ \text{Surface Area} = 2 \cdot 3.14 \cdot 4 , \text{ft} \cdot 7 , \text{ft} \]

3. Calculate step by step:

   • Calculate \( 2 \cdot 3.14 = 6.28 \)
   • Calculate \( 6.28 \cdot 4 = 25.12 \)
   • Calculate \( 25.12 \cdot 7 = 175.84 \)

Thus, the total surface area of the cylinder is:

\[ \text{Surface Area} \approx 175.84 , \text{ft}^2 \]

Now, rounding to the nearest tenth:

\[ \text{Surface Area} \approx 175.8 , \text{ft}^2 \]

所以，Adrian要油漆的圆柱体的表面积是 \( \boxed{175.8} , \text{ft}^2 \)。

Find the volume of the cylinder, in cubic millimeters. Round your answer to the nearest hundredth.
r(8 mm) h(13 mm)

To find the volume \( V \) of a cylinder, we use the formula:

\[ V = \pi r^2 h \]

where:

• \( r \) is the radius,
• \( h \) is the height.

Given:

• Radius \( r = 8 \) mm
• Height \( h = 13 \) mm

Using the approximation \( \pi \approx 3.14 \):

1. Calculate \( r^2 \): \[ r^2 = 8^2 = 64 , \text{mm}^2 \]

2. Substitute \( r^2 \) and \( h \) into the volume formula: \[ V = 3.14 \cdot 64 , \text{mm}^2 \cdot 13 , \text{mm} \]

3. Calculate step by step:

   • First, calculate \( 64 \cdot 13 = 832 \).
   • Then calculate \( 3.14 \cdot 832 \).

   Performing the multiplication: \[ V = 3.14 \cdot 832 = 2611.68 , \text{mm}^3 \]

Therefore, the volume of the cylinder is approximately:

\[ V \approx 2611.68 , \text{mm}^3 \]

Rounding to the nearest hundredth, the final answer is:

\[ \text{Volume} \approx \boxed{2611.68} , \text{mm}^3 \]

Find the surface area of a hemisphere with a radius 10m. Leave your answer in terms of pi.

The surface area \( A \) of a hemisphere can be calculated using the following formula:

\[ A = 2\pi r^2 + \pi r^2 \]

This formula consists of two parts:

1. The curved surface area of the hemisphere, given by \( 2\pi r^2 \).
2. The area of the circular base of the hemisphere, given by \( \pi r^2 \).

So we can simplify this to:

\[ A = 3\pi r^2 \]

Given the radius \( r = 10 , \text{m} \):

1. Calculate \( r^2 \): \[ r^2 = 10^2 = 100 , \text{m}^2 \]

2. Substitute \( r^2 \) into the surface area formula: \[ A = 3\pi (100) = 300\pi \]

Thus, the surface area of the hemisphere is:

\[ \text{Surface Area} = 300\pi , \text{m}^2 \]

Therefore, in terms of \(\pi\), the answer is \( \boxed{300\pi} , \text{m}^2 \).

The surface area of a sphere is 7.84π ft.2
What is the radius of this sphere

To find the radius of a sphere given its surface area, we can use the formula for the surface area \( A \) of a sphere:

\[ A = 4\pi r^2 \]

Given that the surface area \( A = 7.84\pi , \text{ft}^2 \), we can set up the equation:

\[ 4\pi r^2 = 7.84\pi \]

We can divide both sides of the equation by \( \pi \) (assuming \( \pi \neq 0 \)):

\[ 4r^2 = 7.84 \]

Next, we can solve for \( r^2 \):

\[ r^2 = \frac{7.84}{4} = 1.96 \]

Now, take the square root to find \( r \):

\[ r = \sqrt{1.96} = 1.4 , \text{ft} \]

Therefore, the radius of the sphere is \( \boxed{1.4} , \text{ft} \).

Aimee packs ice cream into an ice cream cone. She then puts a perfect hemisphere of ice cream on top of the cone that has a volume of 4 in.3
The diameter of the ice cream cone is equal to its height. What is the total volume of ice cream in and on top of the cone? Use the relationship between the formulas for the volumes of cones and spheres to help solve this problem. Show your work and explain your reasoning.

To find the total volume of ice cream in and on top of the cone, we will first calculate the volume of the cone and then add the volume of the hemisphere (ice cream on top of the cone).

Step 1: Volume of the Hemisphere

The volume \( V \) of a hemisphere is given by the formula:

\[ V = \frac{2}{3} \pi r^3 \]

Given that the volume of the hemisphere is \( 4 , \text{in}^3 \), we can set up the equation:

\[ \frac{2}{3} \pi r^3 = 4 \]

To find \( r^3 \), we can solve for \( r^3 \):

\[ r^3 = \frac{4 \cdot 3}{2\pi} = \frac{12}{2\pi} = \frac{6}{\pi} \]

Now we will find \( r \):

\[ r = \sqrt[3]{\frac{6}{\pi}} \]

Step 2: Diameter and Height of the Cone

From the problem, we know that the diameter of the cone is equal to its height. The diameter \( d \) is twice the radius:

\[ d = 2r \]

Because the height \( h \) of the cone is equal to the diameter, we have:

\[ h = d = 2r \]

Step 3: Volume of the Cone

The volume \( V \) of a cone is given by the formula:

\[ V = \frac{1}{3} \pi r^2 h \]

Since \( h = 2r \):

\[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Step 4: Substitute the Volume of the Hemisphere into the Cone's Volume

We already found that:

\[ r^3 = \frac{6}{\pi} \]

Now substitute \( r^3 \) into the volume of the cone formula:

\[ V = \frac{2}{3} \pi \left(\frac{6}{\pi}\right) = \frac{2 \cdot 6}{3} = 4 , \text{in}^3 \]

Step 5: Total Volume of Ice Cream

The total volume of ice cream in and on top of the cone is the sum of the volume of the cone and the volume of the hemisphere:

\[ \text{Total Volume} = V_{\text{cone}} + V_{\text{hemisphere}} = 4 , \text{in}^3 + 4 , \text{in}^3 = 8 , \text{in}^3 \]

Therefore, the total volume of ice cream in and on top of the cone is:

\[ \boxed{8 , \text{in}^3} \]